Corona .sx The corona formed by a high potential conductor depends , amongst other factors , upon the voltage and upon the radius of curvature of the conductor .sx For a given circuit , therefore , in given physical conditions , the formation of the corona depends upon the value of the voltage .sx As the latter is raised gradually , the corona does not appear until a certain value is reached , after which it continues to grow in magnitude as the voltage is still further raised .sx When the voltage varies very rapidly , however , as during the cycle , the corona does not follow exactly the cyclical variation of the voltage , owing to the fact that the ionization of the air continues for a short time after the electrostatic stress has been removed .sx There is , nevertheless , a certain cyclical change , the corona commencing to form at a certain point on the ascending portion of the voltage wave , reaching a maximum in the near neighbourhood of maximum voltage , and disappearing at a point on the descending portion of the voltage wave .sx This means that the current increases more rapidly when near its maximum value than is the case with a plain unchanging resistance , so that the circuit acts as though it possesses a considerable negative temperature coefficient , being in this respect similar to an arc circuit .sx As the result of the current harmonics , of which the third is the most pronounced , voltage harmonics are also set up , the magnitudes depending upon the constants of the circuit .sx The third harmonic is again very noticeable .sx In an earthed system , this third harmonic voltage causes a triple-frequency current to flow through the capacitance of the system to earth and back through the earthed neutral .sx The general tendency is to peak the current wave and also to introduce harmonics in the voltage wave .sx On occasion the third harmonic current to earth may become very troublesome .sx Pulsating Reactance .sx The main cause of pulsating reactance , apart from the effects of saturation and hysteresis , is due to the cyclical change in reluctance of the magnetic circuit consequent on the synchronous rotation of one part of it .sx Consider , for example , the case of an alternator armature ( it may be either stator or rotor ) in relative motion with its field .sx system .sx In one position the configuration of the iron with respect to the winding may be quite different from what it is in another position .sx With the same current and number of turns the inductance of the winding is then different in different positions .sx With normal symmetrical circuits and machines the conditions repeat themselves after the relative movement through an exact pole pitch , so that two cycles of fluctuation are gone through in every current or voltage cycle .sx The pulsations of reactance are therefore of double frequency , and as a first approximation this reactance may be considered as varying sinusoidally about a mean value .sx When a turn is passing through its zero voltage position it embraces the whole of the useful flux of one pole , and this position may be regarded as the one corresponding to minimum reluctance .sx The flux per ampere and hence the reactance is therefore a maximum in this position .sx When the conductors are passing the centre line of the flux wave , the middle of the turn faces the gap midway between two poles , and the reluctance is greater .sx The inductance corresponding to this position is therefore a minimum .sx Since the current is not necessarily in phase with the voltage , in the general case the phase of the reactance pulsations may be anything with respect to the current .sx The reactance may therefore be represented by .sx where is any angle to take into account the phase of the pulsation .sx This neglects the higher harmonics .sx In the case of the turn considered above , and with unity power factor , is zero .sx Considering first the fundamental component of the current , this gives rise to a reactive voltage of .sx A third harmonic voltage is thus generated in the circuit which , if is zero , goes through its maximum value at the same .sx instant that the reactance is a maximum .sx This harmonic voltage circulates a third harmonic current round the circuit , and this current sets up a third harmonic flux with respect to the armature .sx The latter , however , is moving at synchronous speed with respect to the field , and this would give rise to a flux at fundamental frequency with respect to the armature even if the current were a steady d.c. The combined action of speed and current frequency therefore sets up a flux with respect to the armature proportional to .sx This flux , and hence also the reactance , contains a fourth harmonic .sx In its turn the fourth harmonic flux causes a fifth harmonic voltage to be induced , and this causes a fifth harmonic current to circulate .sx A sixth harmonic flux and a seventh harmonic voltage follow , and the action goes on indefinitely .sx The generation of the fifth harmonic voltage can be predicted in another way , without bringing the fourth harmonic flux into the calculation , except by implication .sx Let the third harmonic current be represented by .sx This causes a p.d. to be set up across the reactance equal to .sx This expression contains , amongst other terms , a fifth harmonic voltage .sx The final result is that the variations in reluctance set up a second harmonic in the reactance ; this engenders a third harmonic current , which in its turn sets up a fourth harmonic in the reactance , and so on .sx All the even harmonics appear in the reactance and all the odd harmonics in the e.m.f. and .sx current .sx The amplitudes of these harmonics gradually die down as the frequency goes up .sx The presence of the harmonics reacts upon the p.d. across any resistance which may be in series , with the usual result that the total p.d. is split up in a more or less unsymmetrical manner between the various component parts of the circuit , harmonics which appear at one part being counterbalanced by other harmonics appearing elsewhere .sx Pulsating Capacitance .sx Certain circuits exhibit effects which are akin to those possessed by a capacitive reactance which pulsates synchronously between upper and lower limits .sx The general behaviour of such circuits is similar to that of circuits possessing pulsating inductive reactance , except that voltage harmonics become magnified in the current wave instead of being damped down .sx One example of a pulsating capacitive reactance is provided by an overhead line on which corona is formed .sx The effective diameter of the conductor for capacitance purposes is the diameter of the corona , and this varies throughout the cycle as the instantaneous value of the voltage changes .sx The capacitance of each of a pair of circular conductors suspended in air is microfarads per mile , where d is the distance between them in cms .sx and r is the radius of each conductor in cms .sx As the corona grows in diameter , d is reduced ( but by a negligible amount ) and r is considerably increased .sx The capacitance is therefore considerably increased .sx The wave of capacitance current is thus peaked , and in certain cases the third harmonic has been found to be as high as 40 per cent .sx of the fundamental .sx As the cyclical changes in capacitance do not obey a simple sine law , the resultant harmonic current is mixed , higher harmonics also being present although the third predominates .sx CHAPTER V. .sx HARMONICS IN POLYPHASE SYSTEMS .sx Three-Phase Line Voltage .sx In a three-phase star-connected system , the line voltage is times the phase voltage and has the same wave-form if the phase voltage is purely sinusoidal .sx If the phase voltage contains harmonics , however , the no longer holds good , and the line voltage wave-form is no longer the same as that of the phase voltage .sx ( Whilst there are theoretical exceptions , this is true in all practical cases .sx ) This is due to the fact that whereas the fundamental line voltage is the difference of two fundamental phase voltages displaced in phase by 120 , the harmonic line voltage is the resultant of harmonic phase voltages at different phase angles .sx The multiplier of holds good for the fundamental components , but different constants have to be employed when the effects of harmonics are taken into consideration .sx In general the nth harmonics in two adjacent phases are out of phase with each other , and the resultant of each pair is their vector difference .sx This resultant voltage is equal to .sx The r.m.s. value of this voltage is .sx The ratio of the line to the phase voltage is , and is shown in the following table for the fundamental and the various harmonics :sx When a third harmonic is present in the phase voltage , its resultant r.m.s. value is , where and are maximum values .sx The r.m.s. value of the line voltage is , since the harmonic voltages cancel out between lines .sx The ratio of the r.m.s. line voltage to the r.m.s. phase voltage is therefore .sx The values of this ratio for third harmonics of various magnitudes are shown in the following table and also in Fig. 55 :sx In a delta-connected system , third harmonic e.m.f.s may exist in the several legs of the delta , but as these harmonics are all in phase with one another , the delta constitutes a short circuit path for them .sx The voltage drop in each leg of the delta is therefore equal to any third harmonic e.m.f. which may exist in the leg , so that no third harmonic voltage exists between lines .sx In any balanced three-phase system , the three lines are equipotential as far as the third harmonic is concerned , so that no third harmonic voltage can exist between them .sx Circuits may be connected between the lines in various manners so as to split up the line voltage into two or more components , and third harmonic voltages may exist across these component parts , but they always total up to zero when taken between line and line .sx Elimination of Third , Ninth , etc. , Harmonics Between Lines .sx In any balanced three-phase three-wire system with insulated neutral , the third and all other harmonics which are a multiple of three , cancel out in the voltage between lines .sx This can be shown in the following manner :sx Let , and be the instantaneous voltages between each line and neutral , considered as acting away from the star point , and assume that these voltages contain harmonics .sx Then .sx .sx The voltage between lines is , , and respectively .sx But , so that the third harmonic disappears .sx In the same way the ninth , fifteenth , twenty-first , etc. , harmonics cancel out , for and , etc. .sx This effect is shown graphically in Fig. 56 , which shows two voltages displaced by 120 , each containing a prominent third harmonic .sx Their difference is a pure sine wave , since no fifth , seventh , etc. , harmonics , entered into their composition .sx Reversal of Fifth Harmonic Between Lines .sx When a fifth harmonic occurs in the phase voltage , it appears in the line voltage with its phase reversed .sx That is , if the harmonic is a peaking one in the phase voltage , it becomes a dimpling or flat-topping one in the line voltage and vice versa .sx The voltage between lines is as before .sx , , .sx But , so that .sx If , then the fundamental and the harmonic reach a positive maximum value simultaneously in the phase voltage , but in the line voltage when one attains a positive maximum value , the other simultaneously attains a maximum value in the reverse direction .sx This effect is shown graphically in Fig. 57 which shows two similar waves with a phase difference of 120 , each containing prominent fifth harmonics of the same magnitude .sx The peaked phase voltages become converted into a dimpled line voltage .sx The relative magnitude of the harmonic , however , is unchanged .sx It is only the phase that is altered .sx Reversal of Seventh Harmonic Between Lines .sx A seventh harmonic in the phase voltage is also reversed when it appears in the line voltage .sx For a seventh harmonic to bring about a peaking effect in the phase voltage , it must be in opposition to the fundamental .sx